There are two questions that I aim to clarify, in light of my own understanding, through this post. The first question is, what is an axiom? An axiom is essentially to math, what a universal truth is to life. For instance, one of these universal truths, though not often thought about or commonly used, is that the truth is always the truth, even if no one believes it. So what this is saying is that, in life, if something happened and is true, even to the knowledge of only a single person, that true thing is, in fact, what actually happened. This holds true whether everyone or no one else thinks that this "true thing" is true. It is something that can be applied to any situation and must be accepted as real or fact by the person to whom it is true. This expands to all of life, in fact.

What this looks like in mathematics is an axiom. So an axiom is a truth about math, that we hold to be true regardless of the situation. In mathematical terms, this truth would in actuality be reminiscent of a proof. Though a proof must be proven to be accepted as true, an axiom (a statement about an aspect of math) is accepted as true without having to prove it....what the heck math!? Anyhow, I'm not sure who, but somewhere along the line some mathematician, who definitely was not bizarre in at least one way, decided that these statements didn't need to be proven. Who'd have thought?

So, what in the world do we use these axioms for? Why would we need something that is accepted as true, without having been proven? Axioms are used for proving other mathematical statements which could not otherwise be proven. That is, for many mathematical statements you come to a point in the proof of said statement where you have to accept at least something as true. If you didn't then you would have to continue breaking down the statement forever...but seriously it would get frustrating or impossible to prove some things if you didn't accept some truth as true. That truth is an axiom. They are essentially the most basic statement that you can use in a proof that doesn't have to be proven itself. An example would be that if we could not assume that all right angles were congruent, we could not logically apply the pythagorean theorem to any triangle. This would be in light of the fact that, if not all right angles are 90 degrees, then they could be other degrees. Smaller or larger. A change in angle measure means a change in segment length on a triangle, which would be impossible to calculate with the pythagorean theorem, because the pythagorean theorem only works if there is a right angle.

That seems confusing. But a simple way to think of an axiom is this. "I would not be at Grand Valley right now if I didn't apply, I wouldn't have applied if I didn't know it existed, I wouldn't have known it existed if...if...if....if...if....................axiom." An axiom is the last in the line of the "ifs"...if you will.

Good explanation of axioms, with colorful support.

ReplyDeleteclear, coherent, complete, consolidated: +

content: while for Euclid the idea of axiom was truth, now we think of it as starting assumption. The classic example being E's own 5th postulate about parallel lines. In a lot of circumstances, that's a good assumption. Not in relativistic physics, it turns out, so we substitute another.

Side note: this is blogpost 1 for weekly work - you still need another for week 2.

Jason, I really like the way you started off this blog post by generalizing the idea of an axiom to everyday life. Students will be able to better grasp this I think before shifting to a more mathematical explanation of the concept. Also, the example at the end with the "I would not be at Grand Valley right now if..." is incredibly creative and comprehensible--even to those who may not be mathematically minded. This is awesome!

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